A temporally and spatially open universe

[Naty1]

In the Glossary of his 1997 book, THE INFLATIONARY UNIVERSE, Alan Guth has this to say:

OPEN UNIVERSE: A homogeneous and isotropic universe is said to be temporally open if gravity is not strong enough to eventually reverse the expansion, so the universde goes on expanding forever. It is said to be spatially open...if triangles contain less than 180 degrees...

OK so far, even though I thought 'open' and 'closed' was usually referring the spatial geometry. anyway, here is the part of interest:

...IF EINSTEIN'S COSMOLOGICAL CONSTANT IS ZERO, AS IS FREQUENTLY ASSUMED, THEN A UNIVERSE WHICH IS TEMPORALLY OPEN IS ALSO SPATIALLY OPEN, AND VICE VERSE.

So I checked to see what he said about a 'closed universe' and sure enough it is consistent:

...IF EINSTEIN'S COSMOLOGICAL CONSTANT IS ZERO, AS IS FREQUENTLY ASSUMED, THEN A UNIVERSE WHICH IS TEMPORALLY CLOSED IS ALSO SPATIALLY CLOSED, AND VICE VERSA.

Is it obvious that open and closed time and space go together? Why only for a zero cosmological constant? And what are the time and space relationships if the cosmological constant isn't exactly zero, say small positive, as is currently thought?

 

[cepheid]

No, it's not meant to be obvious. With non-zero cosmological constant, the universe almost always expands forever (no Big Crunch), and yet it can be spatially open or closed, depending on the values of the density parameters. So the cosmological constant breaks this link between the universe's geometry and its ultimate fate.

As for your question of why they are linked in the first place (for zero cosmo constant), that is not meant to be obvious either, but it can be shown using the Friedmann equations:
http://en.m.wikipedia.org/wiki/Friedmann_equations

In other words, it's a direct result of general relativity (GR). I can explain why the link happens qualitatively. You may recall that GR says the geometry of spacetime is affected by its mass-energy content. For the universe as a whole, this content determines the geometry and the dynamics of the expansion. There is a critical value for the matter density. Above this density of matter, there is enough "stuff" with enough gravity to eventually slow and reverse the universe's expansion, causing it to recollapse (temporally closed). There is also enough stuff to cause a positive spatial curvature (spatially closed -- geometry works like it does on the surface of a sphere). AT the critical density, there is no spatial curvature (neither closed nor open, but "flat" -- ordinary Euclidean geometry), and there isn't enough stuff to slow and stop the expansion. The universe expands forever (temporally open). Below the critical density, the universe also expands forever, and is spatially open (negative curvature -- geometry works like it does on the surface of a "saddle").

The presence of the cosmological constant in the Friedmann equations complicates things. We no longer have these three clean scenarios.

 

[Naty1]

I suspect I did not understand the simplistic meaning of temporally open and closed.

For space, I have seen illustrations, as you describe, of flat and saddle shape open cosmologies, and the closed positive curvature spherical case. None of the curvatures I know a bit about ,like Ricci scalar and Riemann curvature, associated with GR even make mention of time, just space, and often compare those so called 'spacetime' curvatures to the Euclidean case where time is not even part of the geometry. So it seems all the 'curvature' is space like.

So color me surprised if cosmology and GR turns out to have time to have a geometric character ...the same in all the possibilities. Guess I did not think enough about that.

Figure four in the following link is something along the lines I was intuitively expecting...note how time folds around on itself. Analogous to a wormhole maybe.

http://www.rpi.edu/dept/phys/Courses/Astronomy/CurvedSpacetimeAJP.pdf

So time in cosmology seems to be especially simplistic compared to 'curved' space. I am so far unable to see any 'curved time'...yet I know time varies via gravitational potential and relative velocity. Sounds like we never actually attribute 'curvature' to time.

 

[Mordred]

That is a good article, thanks for posting it will be handy on my webpage

 

[Naty1]

Wording got mixed up here:

So color me surprised if cosmology and GR turns out to have time to have a geometric character

it's supposed to read...

So color me surprised if cosmology and GR turns out NOT to have time have a geometric character...

That is a good article, thanks for posting it will be handy on my webpage

sure ...It has some nice illustrations...I got the link from someone in another discussion in these forums...